Remark on Magnetic Schrödinger Operators in Exterior Domains

m at h . SP ] 1 1 Ju l 2 00 9 REMARK ON MAGNETIC SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS

We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulat...

A remark on boundedness of composition operators between weighted spaces of holomorphic functions on the upper half-plane

In this paper, we obtain a sucient condition for boundedness of composition operators betweenweighted spaces of holomorphic functions on the upper half-plane whenever our weights are standardanalytic weights, but they don't necessarily satisfy any growth condition.

A remark on L-boundedness of wave operators for two dimensional Schrödinger operators

Let H = −∆ + V be a two dimensional Schrödinger operator with a real potential V (x) satisfying the decay condition |V (x)| ≤ C〈x〉−δ, δ > 6. Let H0 = −∆. We show that the wave operators s-limt→±∞ eitHe−itH0 are bounded in Lp(R2) under the condition that H has no zero resonances or bound states. In this paper thecondition ∫ R2 V (x)dx 6= 0, imposed in a previous paper (K. Yajima, Commun. Math. P...

Program of a Mini - Course “ Resonances and Threshold Singularities for Magnetic Quantum Hamiltonians ”

1. Basic facts from the spectral theory of magnetic quantum Hamilto-nians (Schrödinger, Pauli, and Dirac operators with magnetic fields): self-adjointness, gauge invariance, diamagnetic inequality, Aharonov-Casher theorem [1, 7]. Constant magnetic fields [1]. 2. Berezin-Toeplitz operators and pseudodifferential operators with con-4. Resonances for the 3D Schrödinger operator with constant magne...

An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Unformly Elliptic Operators with Non–negative Potentials